Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2008, Article ID 632473, pages

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Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 632473, 13 pages
doi:10.1155/2008/632473
Research Article
h-Stability of Dynamic Equations on Time Scales
with Nonregressivity
Sung Kyu Choi,1 Yoon Hoe Goo,2 and Namjip Koo1
1
2
Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea
Department of Mathematics, Hanseo University, Seosan 352-820, South Korea
Correspondence should be addressed to Namjip Koo, njkoo@math.cnu.ac.kr
Received 25 March 2008; Accepted 31 May 2008
Recommended by Allan Peterson
We study the h-stability of dynamic equations on time scales, without the regressivity condition on
the right-hand side of dynamic equations. This means that we can include noninvertible difference
equations into our results.
Copyright q 2008 Sung Kyu Choi et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
The theory of calculus on time scales, which has recently received a lot of attention, was created
by Hilger 1 in order to unify the theories of differential equations and of difference equations
and in order to extend those theories to other kinds of the so-called “dynamic equations.” The
two main features of the calculus on time scales are unification and extension of continuous
and discrete analysis.
Time scale calculus is especially useful when studying systems with discrete and continuous elements in their domains, such as an insect population that is continuous for parts of the
year and discrete for other parts of the year.
The calculus on time scales and dynamic equations on time scales have applications in
any field that requires simultaneous modeling of continuous and discrete processes, because
they bridge the divide between continuous and discrete aspects of processes. The applications
include insect population models, epidemic models, neural networks, and heat transfer.
Foundational definitions and results from the time scale calculus appear in an excellent
introductory text by Bohner and Peterson 2.
In 3, we extended the concept of h-stability introduced by Pinto 4 to dynamic
equations on time scales. The notion of h-stability is quite flexible because it includes the
2
Abstract and Applied Analysis
classical notions of uniform or exponential stability within one common framework. For the
detailed results of h-stability for differential and difference systems, see 4–7.
Regressivity plays a crucial role in developing the fundamental theory of linear dynamic
equations. Consider the scalar dynamic equation on time scale T:
xΔ t atxt.
1.1
Equation 1.1 is said to be regressive if at satisfies
1 atμt/
0
1.2
for all t ∈ Tκ . A system is nonregressive if it is not regressive. When we consider the generalized
time scale exponential function, we need the concept of regressiveness since the exponential
function ea t, t0 is defined only for at satisfying condition 1.2. The continuous dynamical
systems e.g., ordinary differential equations are always regressive since T R has the
graininess function μt ≡ 0. However, nonregressivity is always a possibility in discrete
dynamical systems e.g., difference equations, where the underlying domain consists of a
mixture of discrete and continuous parts. In fact, if there is even one point in T with nonzero
graininess, then nonregressivity is possible 8.
In this paper, we investigate the h-stability for dynamic equations on time scales, with
the nonregressivity condition. Thus, we improve some results in 3.
2. Calculus on time scales
We mention without proof several foundational definitions and theorems, as well as a general
introduction to the theory of time scales in an excellent introductory text by Bohner and
Peterson 2.
Definition 2.1. A time scale T is any nonempty closed subset of the real numbers R.
One assumes throughout that T has the topology that it inherits from the standard
topology on R.
It is also assumed throughout that in T the interval a, b means the set {t ∈ T : a ≤ t ≤ b}
for the points a < b in T. Since a time scale may or may not be connected, one needs the
following concept of jump operators.
Definition 2.2. The functions σ, ρ : T→T defined by
σt inf{s ∈ T : s > t},
ρt sup{s ∈ T : s < t}
2.1
are called the jump operators.
The jump operators σ and ρ allow the classification of points in T in the following way.
Definition 2.3. A nonmaximal element t ∈ T is said to be right-dense if σt t, and right-scattered
if σt > t. Also, a nonminimal element t ∈ T is called left-dense if ρt t, and left-scattered if
ρt < t.
Sung Kyu Choi et al.
3
Definition 2.4. The function μ : T→R defined by μt σt − t is called the graininess function.
If T has a left-scattered maximum m, then Tκ T − {m}. Otherwise Tκ T.
Definition 2.5. A function f : T→R is called differentiable at t ∈ Tκ with (delta) derivative f Δ t ∈ R
if given ε > 0 there exists a neighborhood U of t such that, for all s ∈ U,
σ
f t − fs − f Δ tσt − s ≤ εσt − s,
2.2
where f σ f ◦ σ. If f is delta differentiable for every t ∈ Tκ , then f : T→R is called delta
differentiable on Tκ .
Some basic properties of delta derivatives are as follows 9, 10.
i If f is differentiable at t ∈ Tκ , then
f σ t ft μtf Δ t.
2.3
ii If both f and g are differentiable at t ∈ Tκ , then the product fg is also differentiable
at t ∈ Tκ with
fgΔ t f Δ tgt f σ tg Δ t ftg Δ t f Δ tg σ t.
2.4
Definition 2.6. The function f : T→R is said to be rd-continuous denoted by f ∈ Crd T, R if
i f is continuous at every right-dense point t ∈ T,
ii lims→t− fs exists and is finite at every left-dense point t ∈ T.
Definition 2.7. Let f ∈ Crd T, R. The function g : T→R is called the antiderivative of f on T if it
is differentiable on T and satisfies g Δ t ft for t ∈ Tκ . In this case, one defines
t
fsΔs gt − ga,
t ∈ T.
2.5
a
Some basic properties of delta integral are as follows 2, 9, 10.
Let f, g : T→R be rd-continuous.
i If r, s ∈ T, α, β ∈ R, then
s
s
s
αft βgt Δt α ftΔt β gtΔt.
r
r
2.6
r
ii If t ∈ Tκ , then
σt
t
fτΔτ μtft.
2.7
4
Abstract and Applied Analysis
iii If T R and a, b ∈ T, then
b
ftΔt a
b
2.8
ftdt,
a
where the integral on the right is the usual Riemann integral from calculus.
iv If a, b consists of only isolated points, then
⎧ ⎪
μtft
⎪
⎪
⎪
t∈a,b
⎪
⎪
b
⎨
ftΔt 0
⎪
⎪
a
⎪
⎪
⎪
⎪
μtft
⎩−
if a < b,
if a b,
2.9
if a > b.
t∈b,a
v If T δ Z, where δ > 0, then
⎧
b/δ−1
⎪
⎪
⎪
⎪
fkδδ
⎪
⎪
⎪
ka/δ
⎪
⎪
b
⎨
ftΔt 0
⎪
⎪
a
⎪
⎪ a/δ−1
⎪
⎪
⎪
⎪
fkδδ
⎪
⎩−
if a < b,
if a b,
2.10
if a > b.
kb/δ
3. h-stability
Let Mn R be the set of all n × n matrices over R and the class of all rd-continuous operators
A : T→Mn R denoted by
3.1
Crd T, Mn R .
We consider the linear homogeneous dynamic system
xΔ Atx, x t0 x0 ,
3.2
where A ∈ Crd T, Mn R.
Definition 3.1 see 8. System 3.2 is said to be regressive if
det I μtAt /0
3.3
for all t ∈ Tκ , where I denotes the n × n identity matrix.
It turns out that condition 3.3 is equivalent to having all of the eigenvalues of At
regressive in the sense of 1.2 8.
The norm of an n × n matrix A is defined to be
|A| max Aj ,
3.4
1≤j≤n
where Aj is the jth column of A.
We recall the notion of the transition matrix of the linear dynamic systems without
regressivity.
Sung Kyu Choi et al.
5
Definition 3.2 see 11, Definition 1.3.5. Let τ ∈ T and A ∈ Crd T, Mn R, and assume that
ΦA : {t, τ ∈ T × T : τ ≤ t}→Mn R is an operator with ΦA τ, τ I for all τ ∈ T. Then, the
unique solution ΦA of the IVP
X Δ t AtXt,
Xτ I
3.5
is called the transition matrix of 3.2.
Definition 3.3 see 2, Definition 5.18. Let τ ∈ T and assume that A ∈ Crd T, Mn R is
regressive. The unique solution ΦA t, τ of IVP 3.5 is called the matrix exponential function
of 3.2.
Note that the solution of 3.2 through t0 , x0 can be represented as xt xt, t0 , x0 ΦA t, t0 xt0 . The transition operator has the following properties.
Lemma 3.4 see 11, Theorem 1.3.9. If A ∈ Crd Tκ , Mn R, then there exists the transition matrix
ΦA t, τ of 3.2 which satisfies the following properties:
i ΦA : {t, τ ∈ T × T : τ ≤ t}→Mn R with ΦA τ, τ I for all τ ∈ T;
ii ΦA t, τ ΦA t, sΦA s, τ for all τ ≤ s ≤ t;
iii
t
ΦΔ
A t, τ AtΦA t, τ ∀τ ≤ t,
τ
ΦΔ
A t, τ −ΦA t, στ Aτ ∀στ ≤ t;
3.6
iv
ΦA στ, τ I μτAτ
∀τ ∈ Tκ ;
3.7
if A ∈ Crd Tκ , Mn R is regressive, that is, I μtAt is invertible for all t ∈ Tκ , then
ΦA : T × T→Mn R satisfies (ii) and (iii) for all t, τ ∈ T;
v ΦA t, τ is invertible in Mn R with ΦA t, τ−1 ΦA τ, t for all t, τ ∈ T.
Assume throughout that T is unbounded above.
We recall the definitions about the various types of stability for the solutions of 3.2.
Definition 3.5. The solution x 0 of 3.2 is said to be stable if to any pair of numbers t0 , ε > 0,
there exists a δ δt0 , ε > 0 such that, for any solution xt, t0 , x0 of 3.2, the inequality
|x0 | ≤ δ implies |xt| < ε for all t ≥ t0 ∈ T. A system is said to be stable if all of its solutions are
stable.
Definition 3.6. The solution x 0 of 3.2 is said to be uniformly stable if it is stable and δ does
not depend on t0 .
Definition 3.7. The solution x 0 of 3.2 is said to be asymptotically stable if it is stable and if
there exists a δ0 > 0 such that |x0 | ≤ δ0 implies |xt|→0 as t→∞.
6
Abstract and Applied Analysis
Pinto 4 introduced the notion of h-stability which is an extension of the notions of
exponential stability and uniform Lipschitz stability.
Definition 3.8. System 3.2 is said to be
i an h-system if there exist a positive function h : T→R and a constant c ≥ 1 such that
x t, t0 , x0 ≤ cx0 hth t0 −1 , t ≥ t0 ,
3.8
for |x0 | small enough here ht−1 1/ht;
ii h-stable if system 3.2 is an h-system and the function h is bounded.
Remark 3.9. If ht e−t , then h-stability coincides with exponential stability, and if ht is
constant, then we have uniform Lipschitz stability.
Example 3.10. A linear dynamic system on time scale T with μt ≤ 1/2,
−2
1
Δ
x t xt,
−1 − sin t − 2
3.9
is h-stable 3.
For the various definitions of stability, we refer to 12 and we obtain the following
possible implications for system 3.2 among the various types of stability:
h-stability ⇒ uniform exponential stability ⇒ uniform Lipschitz stability ⇒ uniform stability
3.10
as in 6. The above implications can be proved by the characterization due to Pinto 7, Lemma
1 for the case T R, in terms of the transition matrix for system 3.2.
Now, we consider the linear dynamic system 3.2 without the regressivity condition.
Firstly, we show that stability for solutions of 3.2 is equivalent to boundedness of
solutions.
Theorem 3.11. All solutions of 3.2 are stable if and only if they are bounded for all t ≥ t0 ∈ T.
Proof. Suppose that the solution x 0 of 3.2 is stable. Then, given any ε > 0, there exists a
δ > 0 such that |x0 | < δ implies |xt, t0 , x0 | < ε. However, |xt, t0 , x0 | |ΦA t, t0 x0 | < ε for all
t ≥ t0 ∈ T. Now, let x0 be a vector δ/2ej for j 1, 2, . . . , n. Then,
j δ
ΦA t, t0 x0 x t, t0 < ε,
2
t ≥ t0 , j 1, . . . , n,
3.11
where xj t, t0 is the jth column of ΦA t, t0 . Thus,
ΦA t, t0 maxxj t, t0 < 2ε ,
1≤j≤n
δ
t ≥ t0 .
3.12
Consequently, for any solution xt, t0 , x0 of 3.2,
x t, t0 , x0 ΦA t, t0 x0 < 2ε x0 Mx0 ,
δ
where M 2ε/δ. That is, all solutions of 3.2 are bounded.
Similarly, we can prove the converse.
t ≥ t0 ,
3.13
Sung Kyu Choi et al.
7
In 12, Theorem 2.1, DaCunha obtained the characterization of uniform stability for the
regressive system 3.2. Also, we have the same characterization for the nonregressive system
3.2 in what follows.
Theorem 3.12. Equation 3.2 is uniformly stable if and only if there exists a constant γ > 0 such that
|ΦA t, t0 | ≤ γ,
t ≥ t0 ∈ T,
3.14
where ΦA t, t0 is a transition matrix of 3.2.
Proof. Suppose that 3.2 is uniformly stable. Then, for any given ε > 0, there exists a δ δε >
0 such that t0 ≤ t1 ∈ T and |xt1 | < δ imply |xt| < ε for all t ≥ t1 ∈ T. Thus,
ΦA t, t1 x t1 < ε
∀t ≥ t1 ≥ t0 ∈ T.
3.15
Since xt1 ΦA t1 , t0 x0 can be selected for any t0 and t1 ≥ t0 , let xt0 be a vector δ/2ej for
each j 1, 2, . . . , n. Then,
ΦA t, t1 ΦA t1 , t0 x0 ΦA t, t0 x0 xj t, t0 δ < ε,
2
t ≥ t1 ≥ t0 ,
3.16
where xj t, t0 is the jth column of ΦA t, t0 . Thus,
ΦA t, t0 maxxj t, t0 < γ,
1≤j≤n
3.17
t ≥ t0 ,
where γ 2ε/δ. We see that |ΦA t, t0 | ≤ γ for all t, t0 ∈ T with t ≥ t0 .
Conversely, suppose that there exists a γ > 0 such that |ΦA t, t0 | ≤ γ for all t, t0 ∈ T with
t ≥ t0 . For any t0 and xt0 x0 , the solution of 3.2 satisfies
xt ΦA t, t0 x0 ≤ ΦA t, t0 x0 ≤ γ x0 ,
t ≥ t0 .
3.18
Thus, uniform stability of 3.2 is established.
Pinto 4 gave the characterization of h-system. We obtain the time scale version in what
follows.
Theorem 3.13. Equation 3.2 is an h-system if and only if there exist a positive function h defined on
T and a constant c ≥ 1 such that
ΦA t, t0 ≤ chth t0 −1 ,
t ≥ t0 ∈ T,
3.19
where ΦA t, t0 is a transition matrix of 3.2.
Proof. Suppose that 3.2 is an h-system. Let x0 be a vector δ/2ej for j 1, . . . , n. Then, we
have
x t, t0 , x0 ΦA t, t0 x0 xj t, t0 δ ≤ x0 chth t0 −1 ,
2
where xj t, t0 is the jth column of ΦA t, t0 .
t ≥ t0 , j 1, 2, . . . , n,
3.20
8
Abstract and Applied Analysis
Thus, we obtain
2 ΦA t, t0 ≤ x0 chth t0 −1 chth t0 −1 ,
δ
3.21
t ≥ t0 .
Conversely, we have
x t, t0 , x0 ΦA t, t0 x0 ≤ ΦA t, t0 x0 ≤ chth t0 −1 x0 ,
t ≥ t0 .
3.22
Hence, 3.2 is an h-system.
Remark 3.14. Theorems 3.11, 3.12, and 3.13 also hold for the linear dynamic systems with
regressivity.
Now, we study the h-stability of the nonlinear perturbed dynamic system without the
regressivity condition on the right-hand side of the equation via Gronwall’s inequality and
Bihari’s inequality on time scales.
We consider the perturbed systems of 3.2:
xΔ Atx Ftx,
x t 0 x0 ,
3.23
where A, F ∈ Crd T, Mn R.
We can obtain the following variation-of-constants formula 11.
Lemma 3.15 see 11. The solution xt, t0 , x0 of system 3.24
xΔ t Atxt gt, x,
t ∈ T,
3.24
where A ∈ Crd T, Mn R and g : T × Rn →Rn is rd-continuous in the first argument with gt, 0 0,
with the initial value xt0 x0 is given by
x t, t0 , x0 ΦA t, t0 x0 t
ΦA t, σs g s, xs Δs,
t ≥ t0 ,
3.25
t0
where ΦA t, t0 is a transition matrix of 3.2.
Theorem 3.16. Suppose that 3.2 is h-stable. Then, 3.23 is h-stable if there exists a positive constant
β such that for all t0 ∈ T,
∞
t0
hs FsΔs ≤ β.
h σs
3.26
Proof. Since 3.2 is h-stable, there exist a constant c ≥ 1 and a positive bounded function ht
such that
ΦA t, t0 ≤ chth t0 −1
3.27
Sung Kyu Choi et al.
9
for all t ≥ t0 ∈ T. By Lemma 3.15, the solution xt of 3.23 satisfies
xt ΦA t, t0 x0 t
ΦA t, σs FsxsΔs,
3.28
t ≥ t0 ,
t0
where ΦA t, t0 is a transition matrix of 3.2.
By taking the norms of both sides of 3.28, we have
xt ≤ chth t0 −1 x0 c
t
−1 hth σs FsxsΔs,
t ≥ t0 .
3.29
t0
Dividing by ht > 0 on both sides,
t
xs
xt
x0 hs
≤c c
Δs,
|Fs|
ht
hs
h t0
t0 h σs
3.30
t ≥ t0 .
In view of Gronwall’s inequality on time scales in 9, we obtain
xt
x0 ≤ c echs/hσs|Fs| t, t0 ht
h t0
⎧ t
log 1 μsc hs/h σs Fs
x0
⎪
⎪
Δs
⎪c exp
⎪
⎨ h t0
μs
t0
t
⎪
x0 ⎪
hs ⎪
⎪
c FsΔs
⎩c exp
h t0
t0 h σs
t
x0 hs c ≤ c exp
FsΔs
h t0
t0 h σs
∞
x0 hs ≤ c exp
c Fs Δs
h t0
t0 h σs
x0 ≤ c ecβ
h t0
if μ/
0,
if μ 0
3.31
for all t ≥ t0 ∈ T. Thus,
xt ≤ dx0 hth t0 −1 ,
t ≥ t0 ,
3.32
where d cecβ ≥ 1. Hence, 3.23 is h-stable.
Corollary 3.17. Suppose that 3.2 is h-stable with bounded ht/hσt for each t ∈ T. Then, 3.23
is h-stable if there exists a positive constant β such that for all t0 ∈ T,
∞
t0
FsΔs ≤ β.
3.33
10
Abstract and Applied Analysis
Corollary 3.18. When T R, 3.23 is h-stable if there exists a positive constant β such that for all
t0 ∈ R,
∞
Fsds ≤ β.
3.34
t0
Corollary 3.19. When T δ Z with a positive constant δ, 3.23 is h-stable if there exists a positive
constant β such that for all t0 ∈ δ Z,
∞
hs Fs ≤ β.
hs δ
st0
3.35
We consider the nonlinear perturbed dynamic system
zΔ t Atzt G t, zt ,
3.36
where A ∈ Crd T, Mn R and G is rd-continuous on T × Rn .
We recall the notion of the class H.
if
Definition 3.20. A function w : R →R belongs to the class H
H1 wu is nondecreasing and continuous for u ≥ 0 and positive for u > 0,
H2 there exists a continuous function φ on R with wαu ≤ φαwu for α > 0, u ≥ 0,
H3 limu→0 wu/u exists.
Theorem 3.21. Suppose that 3.2 is h-stable and
Gt, x ≤ Ftw |x| ,
t ≥ t0 ,
3.37
with corresponding multiplier function φ and
where F is positive and rd-continuous, and w ∈ H
z0 ht
h t0 Ft
λt φ
,
h σt z0 h t0
3.38
∞
d W −1 Wc c λsΔs
t0
in [13, Theorem 5.8]. Then, 3.36 is h-stable.
Proof. It follows from Lemma 3.15 that the solution zt of 3.36 is given by
zt ΦA t, t0 z0 t
ΦA t, σs G s, zs Δs,
t ≥ t0 ,
3.39
t0
where ΦA t, t0 is a transition matrix of 3.2.
By h-stability of 3.2, there exists a positive bounded function ht such that
ΦA t, t0 ≤ chth t0 −1
for all t ≥ t0 ∈ T.
The rest of the proof is the same as that of 3, Theorem 2.10.
3.40
Sung Kyu Choi et al.
11
Now, we obtain some results about the h-stability of system 3.36 in the following
corollaries.
Corollary 3.22. Suppose that 3.2 is h-stable with a nondecreasing function ht and
Gt, x ≤ Ftw |x| , t ≥ t0 ,
3.41
with corresponding multiplier function φα α for all α > 0 and
where w ∈ H
∞
d W −1 Wc c FsΔs .
3.42
t0
Then, 3.36 is h-stable.
Corollary 3.23. Suppose that 3.2 is h-stable and
|Gt, x| ≤ Ftw|x|,
t ≥ t0 ∈ T R,
with corresponding multiplier function φ and
where w ∈ H
h t0 Ft
z0 ht
φ
λt ,
htz0 h t0
∞
−1
d W Wc c λsds .
3.43
3.44
t0
Then, 3.36 is h-stable.
Corollary 3.24. Suppose that 3.2 is h-stable and
Gt, x ≤ Ftw |x| ,
t ≥ t0 ∈ T δ Z,
with corresponding multiplier function φ and
where w ∈ H
h t0 Ft
z0 ht
φ
λt ,
ht δz0 h t0
d W −1 Wc cδ
λs .
3.45
3.46
s∈δ Zt0
Then, 3.36 is h-stable.
We examine the property of h-stability for the perturbed dynamic system 3.24 on time
scale T.
Lemma 3.25. Suppose that k ∈ Crd T × R , R is nondecreasing in the second argument x for each
fixed t ≥ t0 with the property
t
t
xt − k s, xs Δs ≤ yt − k s, ys Δs, t ≥ t0 ∈ T,
3.47
t0
t0
for x, y ∈ Crd T, R . If xt0 < yt0 , then xt < yt for all t ≥ t0 ∈ T.
12
Abstract and Applied Analysis
Proof. Suppose that there exists a τ ∈ T with t0 < τ such that xs < ys for t0 ≤ s < τ and
xτ ≥ yτ. Also, we obtain
τ
τ
xτ − k s, xs Δs > yτ − k s, ys Δs, τ ≥ t0 ∈ T.
3.48
t0
t0
This is a contradiction.
If T R and T δ Z with a positive constant δ, then we can obtain the following results
as corollaries of Lemma 3.25.
Corollary 3.26 see 14, Lemma 2.1. Suppose that k ∈ CR × R , R is nondecreasing in the second
argument x for t ≥ t0 with the property
t
t
xt − k s, xs ds ≤ yt − k s, ys ds, t ≥ t0 ∈ R,
3.49
t0
t0
for x, y ∈ CT, R . If xt0 < yt0 , then xt < yt for all t ≥ t0 ∈ R.
Corollary 3.27 see 5, Lemma 9. Suppose that k : δ Z × R →R is nondecreasing in the second
argument x for t ≥ t0 with the property
xt −
t−δ
t−δ
k s, xs δ ≤ yt −
k s, ys δ,
st0
t ≥ t0 ∈ δ Z,
3.50
st0
for x, y : δ Z→R . If xt0 < yt0 , then xt < yt for all t ≥ t0 ∈ δ Z.
Theorem 3.28. Assume that x 0 of 3.2 is uniformly stable. Suppose that
gt, x ≤ ι t, |x| ,
3.51
where ι ∈ Crd T × R , R is strictly increasing in u for each fixed t ∈ Tt0 with ιt, 0 0. Consider the
scalar dynamic equation
uΔ cιt, u, u t0 u0 , t ∈ Tt0 ,
3.52
where c ≥ 1 is a constant. If u 0 of 3.52 is h-stable, then 3.24 is h-stable whenever u0 c|x0 |.
Proof. It follows from Lemma 3.15 that the solution xt of 3.24 is given by
t
xt ΦA t, t0 x0 ΦA t, σs g s, xs Δs, t ∈ Tt0 .
3.53
t0
Since x 0 of 3.2 is uniformly stable, we obtain
t
t
xt ≤ ΦA t, t0 x0 ΦA t, σs g s, xs Δs ≤ cx0 c ι s, xs Δs.
t0
3.54
t0
Thus, we have
xt − c
t
t0
ι s, xs Δs ≤ cx0 ≤ u0 ut − c
t
t0
ι s, us Δs,
t ∈ Tt0 .
3.55
Sung Kyu Choi et al.
13
By Lemma 3.25, we have |xt| < ut for all t ∈ Tt0 . Since u 0 of 3.52 is h-stable, we obtain
xt ≤ c1 u0 hth t0 −1 ≤ c1 cx0 hth t0 −1 Mx0 hth t0 −1 ,
t ∈ Tt0 ,
3.56
where M c1 c ≥ 1 is a constant. This completes the proof.
Acknowledgment
This work was supported by the Korea Research Foundation Grant founded by Korea
Government MOEHRD, KRF-2005-070-C00015.
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